Fugacity coefficient ideal solution

The fugacity coefficient of thesolid solute dissolved in the fluid phase (0 ) has been obtained using cubic equations of state (52) and statistical mechanical perturbation theory (53). The enhancement factor, E, shown as the quantity ia brackets ia equation 2, is defined as the real solubiUty divided by the solubihty ia an ideal gas. The solubiUty ia an ideal gas is simply the vapor pressure of the sohd over the pressure. Enhancement factors of 10 are common for supercritical systems. Notable exceptions such as the squalane—carbon dioxide system may have enhancement factors greater than 10. Solubihty data can be reduced to a simple form by plotting the logarithm of the enhancement factor vs density, resulting ia a fairly linear relationship (52). 

Thus, the fugacity coefficient of species i in an ideal solution is equal to the fugacity coefficient of pure species i in the same physical state as the solution and at the same T and P. 

The residual Gibbs energy and the fugacity coefficient are useful where experimental PVT data can be adequately correlated by equations of state. Indeed, if convenient treatment or all fluids by means of equations of state were possible, the thermodynamic-property relations already presented would suffice. However, liquid solutions are often more easily dealt with through properties that measure their deviations from ideal solution behavior, not from ideal gas behavior. Thus, the mathematical formahsm of excess properties is analogous to that of the residual properties. 

When Eq. (4-282) is applied to XT E for which the vapor phase is an ideal gas and the liquid phase is an ideal solution, it reduces to a veiy simple expression. For ideal gases, fugacity coefficients and are unity, and the right-hand side of Eq. (4-283) reduces to the Poynting factor. For the systems of interest here this factor is always veiy close to unity, and for practical purposes <1 = 1. For ideal solutions, the activity coefficients are also unity. Equation (4-282) therefore reduces to... 

Activity coefficients are equal to 1.0 for an ideal solution when the mole fraction is equal to the activity. The activity (a) of a component, i, at a specific temperature, pressure and composition is defined as the ratio of the fugacity of i at these conditions to the fugacity of i at the standard state [54]. 

Phase Equilibria Models Two approaches are available for modeling the fugacity of a solute in a SCF solution. The compressed gas approach includes a fugacity coefficient which goes to unity for an ideal gas. The expanded liquid approach is given as... 

Activity ax is termed the rational activity and coefficient yx is the rational activity coefficient This activity is not directly given by the ratio of the fugacities, as it is for gases, but appears nonetheless to be the best means from a thermodynamic point of view for description of the behaviour of real solutions. The rational activity corresponds to the mole fraction for ideal solutions (hence the subscript x). Both ax and yx are dimensionless numbers. 

K is equilibrium constant. If we consider the reference standard state to be gaseous, K = K because the activity and fugacity coefficients are unity in very dilute solution and ideal gas, respectively. Then ( g)0 and (ks)o will be the same. 

Standard states are usually chosen2 so that the activity reduces to the pressure for gases at low pressures, and to concentrations in dilute solution. The choices are summarized in Table 11.2.aa We note that for a gas, the standard state is the ideal gas at a pressure of 1 bar (0.1 MPa), in which case, the activity differs from the pressure (expressed in bars) by the fugacity coefficient. That is,... 

The solubility data for naphthalene in ethylene and in CO2 are consistent with the data in Figure 3. The proper way to make the comparison is to use the enhancement factor instead of the solubility. The enhancement factor equals y2P/P2 which is simply the actual solubility divided by the solubility in an ideal gas. The enhancement factor removes the effect of vapor pressure which is useful for comparing fluids at constant reduced temperature but at different actual temperatures. In terms of the fugacity coefficient of the solute, 2, the enhancement factor is given by... 

Because of the complex functionality of the K-values, these calculations in general require iterative procedures suited only to computer solution. However, in the case of mixtures of light hydrocarbons, in which the molecular force fields are relatively weak and uncomplicated, we may assume as a reasonable approximation that both the liquid and the vapor phases are ideal solutions. By definition of the fugacity coefficient of a species in solution, =fffxtP. But by Eq. (11.61), f f = xj,. Therefore... 

The appropriate expression for the equilibrium equation is Eq. (15.23). equation requires evaluation of the fugacity coefficients of the species prese equilibrium. Although the generalized correlation of Sec. 11.4 is applicable calculations involve iteration, because the fugacity coefficients are functions of sition. For purposes of illustration, we carry out only the first iteration, based assumption that the reaction mixture is an ideal solution. In this case Eq. (1 reduces to Eq. (15.24), which requires fugacity coefficients of the pure reacting at the equilibrium T and P. Since v = = -1, this equation becomes... 

In this equation is the fugacity coefficient of pure saturated i (either H vapor) evaluated at the temperature of the system and at A, the vapor pr pure i. The assumption that the vapor phase is an ideal solution allows sub of < CiH4 for < csh,. where 4>cth s tit fugacity coefficient of pure ethylene system T and P. With this substitution and that of Eq. (F), Eq. ( ) becomes... 

In this definition, the activity coefficient takes account of nonideal liquid-phase behavior for an ideal liquid solution, the coefficient for each species equals 1. Similarly, the fugacity coefficient represents deviation of the vapor phase from ideal gas behavior and is equal to 1 for each species when the gas obeys the ideal gas law. Finally, the fugacity takes the place of vapor pressure when the pure vapor fails to show ideal gas behavior, either because of high pressure or as a result of vapor-phase association or dissociation. Methods for calculating all three of these follow. 

Related Calculations. If the gas is not ideal, the fugacity coefficients , will not be unity, so the activities cannot be represented by the mole fractions. If the pressure is sufficient for a nonideal solution to exist in the gas phase, , will be a function of y, the solution to the problem. In this case, the y, value obtained for the solution with Lewis-Randall rule for... 

A and Tj are the activity coefficient matrices for vapor and liquid phases and are diagonal. For ideal solutions, they become the identity matrix. K is the fugacity ratio matrix and is also diagonal. 

Although the composition of an ideal solution can be predicted theoretically, few solutions are ideal, and fugacities and activity coefficients are seldom available for real systems. Hence, in general, too little is known of the direct relationships between solubilities and the specific properties of the solute and the solvent to permit prediction of solubility curves. The characteristics of each system must be determined experimentally. In many cases, it is not even possible to predict the effect of temperature on the solubility values of a given solute-solvent system. 

If the assumption is justified that the equilibrium mixture is an ideal solution, then each becomes 4>i, the fugacity coefficient of pure i at T and P. In this case, Eq. (15.23) becomes... 

An initial estimate of the vapor-phase composition is based on the assumption that both the liquid and vapor phases are ideal solutions. Each fugacity coefficient is then given by... 

Solubilitiesattemperaturesand pressures above the critical values of the solvent liave important applications for supercritical separation processes. Examples are extraction of caffeine from coffee beans and separation of asplraltenes from heavy petroleum fractions. For a typical solid/vapor equilibrium (SVE) problem, tire solid/vapor saturation pressure P is very small, and the saturated vapor is for practical purposes an ideal gas. Hence 0 for pure solute vapor at this pressure is close to unity. Moreover, exceptfor very low values of the system pressure P, the solid solubility yj is small, and can be approximated by j, the vapor-phase fugacity coefficient of the solute at infinite dilution. Finally, since is very small, the pressure difference P — in the Poyntingfactor is nearly equal to P at any pressure where tins factor... 

Henry s law constants, as other thermodynamic constants, are valid for ideal solutions ideally, the expression should be written in terms of activities and fugacities. Since activity coefficients for uncharged species are much smaller than those for ions, we can use expressions such as equation 3 for diluhj solutions (fresh water) and atmospheric pressures. However, corrections anj necessary for seawater and concentrated solutions. Since activity coefficients for molecules in aqueous solution become larger than 1 (salting out effect), the solubility of gases in concentration units is smaller in the salt solution than in the dilute aqueous medium. 

Eqs. (l)-(3) show that the solubilities of sohds in an SCF depend among others on their fugacity coefficients (pf (pf and the calculations indicated that these coefficients were responsible for the large solubilities of solids in supercritical solvents. These solubilities are much larger than those in ideal gases, and enhancement factors of 10" -10 are not uncommon [1] they are, however, still relatively small and usually do not exceed several mole percent. Consequently, these supercritical solutions can be considered dilute and the expressions for the fugacity coefficients in binary and ternary supercritical mixtures simplihed accordingly. 

Since then further progress has extended the field of applicability of Gibbs chemical thermodynamics. Thus the introduction of the ideas of fugacity and activity by G. N. Lewis enabled the thermodynamic description of imperfect gases and of real solutions to be expressed with the same formal simplicity as that of perfect gases and ideal solutions. These results were completed when N. Bjerrum and E. A. Guggenheim introduced osmotic coefficients. 

By equation (31.5), the activity of the solvent is equivalent to fi/fi where fi is the fugacity in a given solution and / is numerically equal to that in the standard state, i.e., pure liquid at 1 atm. pressure at the given temperature. Hence, it is seen from equation (34.1) that for an ideal solution the activity of the solvent should always be equal to its mole fraction, provided the total pressure is 1 e m. In other words, in these circumstances the activity coefficient ui/ni should be inity at all concentrations. For a nonideal solution, therefore, the deviation of ai/Ni from unity at 1 atm. pressure may be taken as a measure of the departure from ideal (Raoult law) behavior. Since the activities of liquids are not greatly affected by pressure, this conclusion may be accepted as generally applicable, provided the pressure is not too high. 

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